I know that they are tapered style, but I just talked with a shop selling some lug nuts, and they said there are 2 sizes, conventional and a smaller sized lug nut.

 

The K-1s I got from Tire Rack came with 17mm (socket size) closed-end lugs, where the normal size is 19mm. Don't know if that's the smaller size he was telling you about. From what I recall, though, there should have been plenty of room for a 19mm nut, so I would think that a normal size should be fine. Don't have the wheels anymore else I'd go check.

When I got track wheels for the new car, I got lug nuts from AutoZone, and they had a great selection; all different sizes and styles. If you have one nearby, I'd definitely recommend them.

 

Tire Rack normally ships a smaller 17mm lug with the Kosei's in Honda fitments. However, a 19 mm fits fine with plenty of socket clearance, as that's what I use on mine. Actually, since I purchased my K1's back in May, and I made a bit of a stink about the smaller size, supposedly TR has updated their info for Honda fitments.

Good customer service from TR, though. They sent me two sets of additional 19mm lugs.